Answer

Verified

413.1k+ views

**Hint:**We start solving the problem by expanding the given multiplication and then find the terms for which we need to find the coefficient. We then recall the fact that the coefficient of $ {{x}^{r}} $ in the binomial expansion of $ {{\left( a+bx \right)}^{n}} $ , $ \left( n\ge r \right) $ as $ {}^{n}{{C}_{r}}{{a}^{n-r}}{{b}^{r}} $ to find the coefficients of required terms. We then add those obtained coefficients and make the necessary calculations to get the required answer.

**Complete step by step answer:**

According to the problem, we are asked to find the coefficient of $ {{x}^{n}} $ in the expansion of $ \left( 1+x \right){{\left( 1-x \right)}^{n}} $ .

Now, we have $ \left( 1+x \right){{\left( 1-x \right)}^{n}}={{\left( 1-x \right)}^{n}}+x{{\left( 1-x \right)}^{n}} $ ---(1).

In order to find the coefficient of $ {{x}^{n}} $ for the expansion in equation (1), We need to find the coefficient of $ {{x}^{n}} $ for $ {{\left( 1-x \right)}^{n}} $ and $ {{x}^{n-1}} $ for $ {{\left( 1-x \right)}^{n}} $ to get required coefficient in $ x{{\left( 1-x \right)}^{n}} $ .

Let us recall the coefficient of $ {{x}^{r}} $ in the binomial expansion of $ {{\left( a+bx \right)}^{n}} $ , $ \left( n\ge r \right) $ as $ {}^{n}{{C}_{r}}{{a}^{n-r}}{{b}^{r}} $ .

Now, the co-efficient of $ {{x}^{n}} $ in the expansion of $ {{\left( 1-x \right)}^{n}} $ is $ {}^{n}{{C}_{n}}{{\left( 1 \right)}^{n-n}}{{\left( -1 \right)}^{n}} $ .

We know that $ {}^{n}{{C}_{n}}=1 $ . So, the co-efficient of $ {{x}^{n}} $ in the expansion of $ {{\left( 1-x \right)}^{n}} $ is $ {{\left( -1 \right)}^{n}} $ ---(2).

Now, the co-efficient of $ {{x}^{n-1}} $ in the expansion of $ {{\left( 1-x \right)}^{n}} $ is $ {}^{n}{{C}_{n-1}}{{\left( 1 \right)}^{n-n+1}}{{\left( -1 \right)}^{n-1}} $ .

We know that $ {}^{n}{{C}_{n-1}}=n $ . So, the co-efficient of $ {{x}^{n-1}} $ in the expansion of $ {{\left( 1-x \right)}^{n}} $ is $ {{\left( -1 \right)}^{n-1}}n $ ---(3).

Let us use the results obtained in equations (2) and (3) to get the coefficient of $ {{x}^{n}} $ in the expansion of $ \left( 1+x \right){{\left( 1-x \right)}^{n}} $ .

So, the required co-efficient is $ {{\left( -1 \right)}^{n}}+{{\left( -1 \right)}^{n-1}}n={{\left( -1 \right)}^{n}}+\dfrac{{{\left( -1 \right)}^{n}}}{\left( -1 \right)}n={{\left( -1 \right)}^{n}}\left( 1-n \right) $ .

We have found the coefficient of $ {{x}^{n}} $ in the expansion of $ \left( 1+x \right){{\left( 1-x \right)}^{n}} $ as $ {{\left( -1 \right)}^{n}}\left( 1-n \right) $ .

$\therefore$ The correct option for the given problem is (b).

**Note:**

We should not confuse while finding the binomial coefficients of required terms in this problem. Whenever we get this type of problem, we first try to find the required terms whose coefficients we need to find for solving the problem. Similarly, we can expect problems to find the coefficient of $ {{x}^{n}} $ in the binomial expansion of $ \left( 1+x \right){{\left( 1-x \right)}^{-n}} $ .

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Which are the Top 10 Largest Countries of the World?

Difference Between Plant Cell and Animal Cell

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE